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This work is concerned with quasi-optimal a-priori finite element errorestimates for the obstacle problem in the $L^2$-norm. The discreteapproximations are introduced as solutions to a finite element discretizationof an accordingly regularized problem. The underlying domain is only assumed tobe convex and polygonally or polyhedrally bounded such that an application ofpoint-wise error estimates results in a rate less than two in general. The mainingredient for proving the quasi-optimal estimates is the structural andcommonly used assumption that the obstacle is inactive on the boundary of thedomain. Then localization techniques are used to estimate the global$L^2$-error by a local error in the inner part of the domain, where higherregularity for the solution can be assumed, and a global error for theRitz-projection of the solution, which can be estimated by standard techniques.We validate our results by numerical examples.

Finite element error estimates in $L^2$ for regularized discrete approximations to the obstacle problem